**Instructor:**Olga Plamenevskaya, office 3-107 Math Tower, e-mail:`olga@math.sunysb.edu`**Office hours:**MW 12:45am-2:15pm or by appointment.**Class meetings:**Monday and Wednesday, 2:20-3:40pm, Earth&Space 177.

The goal of this course is to provide a basic introduction into singular homology and cohomology.

** Important: ** Please
write up your solutions neatly, be sure to put your name on them and staple all pages.
Illegible homework will not be graded. Late homework will not be accepted.
You are welcome to collaborate with others and even to consult books,
but your solutions should be written up in your own words,
and all your
collaborators and sources should be listed.

- Basic constructions
- Homotopies & homotopy equivalences, homotopy type; retractions, deformation retractions
- CW-complexes, definition, examples; simplicial complexes
- Operations on spaces: products, quotients, wedge sums, suspension, etc

- Homology
- Singular homology & simplicial homology, constructions
- Chain complexes, chain maps, chain homotopies; exact sequences; the Euler characteristic
- Homotopy invariance of singular homology
- Relative homology, long exact sequence of a pair
- The excision theorem
- Equivalence of simplicial and singular homology
- Homology of CW-complexes via cellular homology
- Computations: surfaces, spheres, projective spaces, lens spaces, etc
- Mayer-Vietoris sequence, more calculations
- Applications: Brouwer fixed point thm, degrees, Jordan curve thm, invariance of domain, etc
- Eilenberg-Steenrod axioms

- Cohomology
- Simplicial and singular cohomology groups
- Hom and Ext, universal coefficient theorems
- Relative cohomology, exact sequences, isomorphism between simplicial and singular cohomology
- Cup product, calculations (cohomology ring of projective spaces, etc)
- Kunneth formulas
- Poincare duality

**Students with Disabilities: **If you have a physical,
psychological, medical, or learning disability that may impact on your
ability to carry out assigned course work, you are strongly urged to
contact the staff in the Disabled Student Services (DSS) office: Room
133 in the Humanities Building; 632-6748v/TDD. The DSS office will
review your concerns and determine, with you, what accommodations are
necessary and appropriate. A written DSS recommendation should be
brought to your lecturer who will make a decision on what special
arrangements will be made. All information and documentation of
disability is confidential. Arrangements should be made early in the
semester so that your needs can be accommodated.